Control-Oriented Reduced-Order Models

Control-Oriented Reduced-Order Models for Lithium-Metal Batteries

22nd Advanced Automotive Battery Conference– San Diego, CA – December 5-8, 2022

Authors: Aloisio Kawakita de Souza, Wesley Hileman, M. Scott Trimboli, Gregory L. Plett

Department of Electrical and Computer Engineering, University of Colorado Colorado Springs

Background

Lithium-metal batteries (LMB) hold promise as successors to lithium-ion batteries (LIB) due to its high energy-density.
LMB have metallic lithium anodes which introduce complications to modelling their long-term behaviour. Particularly, a dead-lithium layer grows over time, and this must be described in any battery-management-system (BMS) model to enable accurate estimates of state-of-charge, state-of-health, and power limits.

Objectives

Develop two physics-based control-oriented models of LMBs for the design of advanced BMS algorithms:

A reduced-order discrete-time model generated via a subspace-based method and the linearized transfer functions derived from the fundamental equations.
A simplified model based on polynomial functions that compute the solid diffusion in the positive electrode and the electrolyte dynamics across the cell.

Pseudo Two-Dimensional Model

The pseudo two-dimensional (P2D) model adopted from [1] describes the processes for the electrolyte phase in the 𝑥 dimension across the cell sandwich. While in the positive solid electrode, the porous structure is represented by spherical particles with radial dimension 𝑟 at each channel location in 𝑥.

Moreover, Xu et al. [1] models the Li metal negative electrode as an electrode surface with a high electrical conductivity and includes a static description of the dead-lithium layer at a particular state-of-age of a LMB cell. It does not specify how the dead-lithium layer evolves as a cell ages.
This model is the foundation of our work, but it is too computationally complex to be used directly in real-time BMS algorithms.

Enhanced Single Particle Model

The single particle model (SPM) is based mainly on the two following assumptions [2]:
- [A1] The positive electrode is assumed to be a single spherical particle whose surface area is equivalent to the active area of the porous electrode.
- [A2] The total moles of lithium in the electrolyte and in the solid phase are both conserved.

The assumptions above implicate that:
- (a) The lithium concentration in the solid phase does not depend on spatial distance 𝑥 across the electrode.
- (b) The intercalation reaction flux is assumed to be proportional to the applied current.
The electrolyte dynamics is incorporated to the SPM to improve voltage prediction at high C-rate applications.
Polynomial functions are used to approximate the solution of the solid diffusion and electrolyte concentration resulting in a linear state-space system [3].

Physics-Based Reduced-Order Model

The physics-based reduced-order model (ROM) is obtained by a transfer-function based approach that generates linear state-space (SS) models at different SOC and temperature setpoints.

The realization algorithms (xRAs) optimizes the error between the approximate model and the discrete-time frequency response of the electrochemical variables for a specific 𝑥 location of interest.

The linear SS models are blended together at every time step using a bilinear interpolation scheme between the four closest models to the present SOC and temperature.

Simulation Results

Simulation results for the ROM and SPM are compared against a virtual cell implemented in COMSOL using the cell parameters from [1].
Case 1: Constant current discharge
- Cell is discharged from 90% to 5% SOC at 0.5, 1, 2 and 3 C-rates.

Case 2: Dynamics driving profile
- Cell is discharged from initial SOC of 80 % using an Urban Dynometer Driving Schedule (UDDS)

Highlights

Physics-based control-oriented models have been developed for LMBs based on previous work on LIBs.
The simulation results for ROM and SPM are compared against a virtual cell implemented in COMSOL via FOM using the cell parameters from [1].
ROM has shown superior performance over the SPM for constant current discharge profiles above 2 C-rate.
Both ROM and SPM demonstrated accurate voltage predictions for dynamics driving profiles
Future work will include temperature dynamics, aging and experimental results.

References

S. Xu, K.-H. Chen, N. Dasgupta, J. Siegel, and A. Stefanopoulou, “Evolution of dead lithium growth in lithium metal batteries: Experimentally validated model of the apparent capacity loss,” Journal of The Electrochemical Society, vol. 166, pp. A3456–A3463, 01 2019.
S. J. Moura, F. B. Argomedo, R. Klein, A. Mirtabatabaei, and M. Krstic, “Battery State Estimation for a Single Particle Model with Electrolyte Dynamics,” IEEE Trans Contr Syst Technol, vol. 25, no. 2, pp. 453–468, Mar.
Rahimian, S. K., Rayman, S., & White, R. E. “Extension of Physics-Based Single Particle Model for Higher Charge-Discharge Rates. Journal of Power Sources”, 224, 180 – 194.
Lu, D. “Identifying Physical Model Parameter Values for Lithium-Ion Cells”. Ph.D. Thesis, University of Colorado Colorado Springs, Colorado Springs, CO, USA, 2022.